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Maps Preserving Complementarity of Closed Subspaces of a Hilbert Space

Published:2013-08-20

• Lucijan Plevnik,
Institute of Mathematics, Physics, and Mechanics, Jadranska 19, SI-1000 Ljubljana, Slovenia
• Peter Šemrl,
Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia
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Abstract

Let $\mathcal{H}$ and $\mathcal{K}$ be infinite-dimensional separable Hilbert spaces and ${\rm Lat}\,\mathcal{H}$ the lattice of all closed subspaces oh $\mathcal{H}$. We describe the general form of pairs of bijective maps $\phi , \psi : {\rm Lat}\,\mathcal{H} \to {\rm Lat}\,\mathcal{K}$ having the property that for every pair $U,V \in {\rm Lat}\,\mathcal{H}$ we have $\mathcal{H} = U \oplus V \iff \mathcal{K} = \phi (U) \oplus \psi (V)$. Then we reformulate this theorem as a description of bijective image equality and kernel equality preserving maps acting on bounded linear idempotent operators. Several known structural results for maps on idempotents are easy consequences.
 Keywords: Hilbert space, lattice of closed subspaces, complemented subspaces, adjacent subspaces, idempotents
 MSC Classifications: 46B20 - Geometry and structure of normed linear spaces 47B49 - Transformers, preservers (operators on spaces of operators)